标题： 通过仿射簇维度理解物理学信息模型的泛化性 显示英文标题

标题： Understanding Generalization in Physics Informed Models through Affine Variety Dimensions

摘要： 近年来，物理信息机器学习因其能够通过将物理结构整合到机器学习模型中来提升统计性能和样本效率而引起了广泛关注。这些结构，例如微分方程、守恒定律和对称性，作为归纳偏置可以提高混合模型的泛化能力。然而，这些物理结构如何增强泛化能力的机制尚未被完全理解，这限制了对模型性能保证的能力。在这项研究中，我们表明，包含微分方程结构的线性回归器的泛化性能由相关仿射簇的维度决定，而不是参数的数量。这一发现使我们可以对各种方程（包括非线性方程）进行统一分析。我们引入了一种方法来近似计算仿射簇的维度，并提供了实验证据以验证我们的理论见解。 显示英文摘要

摘要： In recent years, physics-informed machine learning has gained significant attention for its ability to enhance statistical performance and sample efficiency by integrating physical structures into machine learning models. These structures, such as differential equations, conservation laws, and symmetries, serve as inductive biases that can improve the generalization capacity of the hybrid model. However, the mechanisms by which these physical structures enhance generalization capacity are not fully understood, limiting the ability to guarantee the performance of the models. In this study, we show that the generalization performance of linear regressors incorporating differential equation structures is determined by the dimension of the associated affine variety, rather than the number of parameters. This finding enables a unified analysis of various equations, including nonlinear ones. We introduce a method to approximate the dimension of the affine variety and provide experimental evidence to validate our theoretical insights.